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# C8D2 - 1 We will now look at the properties of the OLS...

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Unformatted text preview: 1 We will now look at the properties of the OLS regression estimators with the assumptions of Model B. We will do this within the context of the simple regression model. We will start by demonstrating unbiasedness. ( 29 ( 29 ( 29 ∑ ∑ ∑ + =--- = i i i i i u a X X Y Y X X b 2 2 2 β i i i u X Y + + = 2 1 β β MODEL B: PROPERTIES OF THE REGRESSION COEFFICIENTS 2 We have seen that the slope coefficient can be decomposed into the true value plus a weighted linear combination of the values of the disturbance term in the sample, where the weights depend on the observations on X . ( 29 ( 29 ( 29 ∑ ∑ ∑ + =--- = i i i i i u a X X Y Y X X b 2 2 2 β i i i u X Y + + = 2 1 β β ∑-- = 2 ) ( ) ( X X X X a i i i MODEL B: PROPERTIES OF THE REGRESSION COEFFICIENTS 3 We now take expectations. β 2 is just a constant, so it is unaffected. ( 29 ( 29 ( 29 ∑ ∑ ∑ + =--- = i i i i i u a X X Y Y X X b 2 2 2 β i i i u X Y + + = 2 1 β β ∑-- = 2 ) ( ) ( X X X X a i i i ( 29 ( 29 ( 29 ∑ ∑ + = + = i i i i u a E u a E b E 2 2 2 β β MODEL B: PROPERTIES OF THE REGRESSION COEFFICIENTS 4 We have now used the first expectation rule to rewrite the expectation of the linear combination as the sum of the expectations of its components. ( 29 ( 29 ( 29 ∑ ∑ ∑ + =--- = i i i i i u a X X Y Y X X b 2 2 2 β i i i u X Y + + = 2 1 β β ∑-- = 2 ) ( ) ( X X X X a i i i ( 29 ( 29 ( 29 ∑ ∑ + = + = i i i i u a E u a E b E 2 2 2 β β ( 29 ( 29 ( 29 ( 29 ( 29 ∑ ∑ = + + = + + = i i n n n n i i u a E u a E u a E u a u a E u a E ... ... 1 1 1 1 MODEL B: PROPERTIES OF THE REGRESSION COEFFICIENTS 5 In Model A, the values of X were nonstochastic. This meant that the a i terms were also nonstochastic and could therefore be taken out of the expectations as factors. E ( u i ) = 0 for all i , and hence we proved unbiasedness. ( 29 ( 29 ( 29 ∑ ∑ ∑ + =--- = i i i i i u a X X Y Y X X b 2 2 2 β i i i u X Y + + = 2 1 β β ∑-- = 2 ) ( ) ( X X X X a i i i ( 29 ( 29 ( 29 ∑ ∑ + = + = i i i i u a E u a E b E 2 2 2 β β ( 29 ( 29 2 2 2 : A Model β β = + = ∑ i i u E a b E MODEL B: PROPERTIES OF THE REGRESSION COEFFICIENTS 6 ( 29 ( 29 ( 29 ∑ ∑ ∑ + =--- = i i i i i u a X X Y Y X X b 2 2 2 β i i i u X Y + + = 2 1 β β ∑-- = 2 ) ( ) ( X X X X a i i i ( 29 ( 29 ( 29 ∑ ∑ + = + = i i i i u a E u a E b E 2 2 2 β β ( 29 ( 29 2 2 2 : A Model β β = + = ∑ i i u E a b E We cannot do this with Model B because we are assuming that the values of X are generated randomly (from a defined population)....
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C8D2 - 1 We will now look at the properties of the OLS...

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